 The above work yields an expression for \(G_{\partial M}\)  centred at the origin of the window \(x^*\in \Gamma _{\varepsilon ,a}\) . We however need an expansion for \(G_{\partial M}(x,y)\)  for \(x,y\)  close in \(\Gamma _{\varepsilon ,a}\) . In particular, we need an expression given by (REF ). In order to do so, we simply invoke the following proposition, which was proven in [1]}.
